Turán number conjecture for Cartesian products of trees

Let $T$ and $S$ be trees, each with at least one edge, and let $T\Box S$ denote their Cartesian product. For a positive integer $n$, let $\mathrm{ex}(n,T\Box S)$ be the maximum number of edges in a $(T\Box S)$-free graph on $n$ vertices. Cartesian-product conjecture. There exist positive real numbers $c$ and $C$ such that

$$cn^{3/2}\leq \mathrm{ex}(n,T\Box S)\leq Cn^{3/2}.$$

The product $T\Box S$ is $2$-degenerate, and the conjecture is motivated in part by Erdős's conjecture on Turán numbers of degenerate bipartite graphs. Its general validity remains open.